|To inscribe a square in a given circle.|
|Let ABCD be the given circle.
It is required to inscribe a square in the circle ABCD.
|Draw two diameters AC and BD of the circle ABCD at right angles to one another, and join AB, BC, CD, and DA.||III.1
|Then, since BE equals ED, for E is the center, and EA is common and at right angles, therefore the base AB equals the base AD.||I.4|
|For the same reason each of the straight lines BC and CD also equals each of the straight lines AB and AD. Therefore the quadrilateral ABCD is equilateral.
I say next that it is also right-angled.
|For, since the straight line BD is a diameter of the circle ABCD, therefore BAD is a semicircle, therefore the angle BAD is right.||III.31|
|For the same reason each of the angles ABC, BCD, and CDA is also right. Therefore the quadrilateral ABCD is right-angled.|
|But it was also proved equilateral, therefore it is a square, and it has been inscribed in the circle ABCD.|
|Therefore the square ABCD has been inscribed in the given circle.|
Next proposition: IV.7